1, A brief review about Quantum Mechanics

Transcription

1 IPT5340, Spring 08 p. 1/5 1, A brief review about Quantum Mechanics 1. Basic Quantum Theory 2. Time-Dependent Perturbation Theory 3. Simple Harmonic Oscillator 4. Quantization of the field 5. Canonical Quantization Ref: Ch. 2 in Introductory Quantum Optics, by C. Gerry and P. Knight. Ch. 2 in Mesoscopic Quantum Optics, by Y. Yamamoto and A. Imamoglu. Ch. 1 in Quantum Optics, by D. Wall and G. Milburn. Ch. 4 in The Quantum Theory of Light, by R. Loudon. Ch. 1, 2, 3, 6 in Mathematical Methods of Quantum Optics, by R. Puri Ch. 3 in Elements of Quantum Optics, by P. Meystre and M. Sargent III. Ch. 9 in Modern Foundations of Quantum Optics, by V. Vedral.

2 IPT5340, Spring 08 p. 2/5 Field Quantization 1. Simple Harmonic Oscillator 2. Quantization of a single-mode field 3. Basic Quantum Theory 4. Time-Dependent Perturbation Theory 5. Canonical Quantization 6. Quantum fluctuations of a single-mode field 7. Quadrature operators for a single-mode field 8. Multimode fields 9. Thermal fields 10. Vacuum fluctuations and the zero-point energy 11. Casimir force

3 IPT5340, Spring 08 p. 3/5 Role of Quantum Optics photons occupy an electromagnetic mode, we will always refer to modes in quantum optics, typically a plane wave; the energy in a mode is not continuous but discrete in quanta of ω; the observables are just represented by probabilities as usual in quantum mechanics; there is a zero point energy inherent to each mode which is equivalent with fluctuations of the electromagnetic field in vacuum, due to uncertainty principle. quantized fields and quantum fluctuations (zero-point energy)

4 IPT5340, Spring 08 p. 4/5 Vacuum vacuum is not just nothing, it is full of energy.

5 IPT5340, Spring 08 p. 5/5 Vacuum spontaneous emission is actually stimulated by the vacuum fluctuation of the electromagnetic field, one can modify vacuum fluctuations by resonators and photonic crystals, atomic stability: the electron does not crash into the core due to vacuum fluctuation of the electromagnetic field, gravity is not a fundamental force but a side effect matter modifies the vacuum fluctuations, by Sakharov, Casimir effect: two charged metal plates repel each other until Casimir effect overcomes the repulsion, Lamb shift: the energy level difference between 2S 1/2 and 2P 1/2 in hydrogen....

6 IPT5340, Spring 08 p. 6/5 Simple Harmonic Oscillator The simple harmonic oscillator has no driving force, and no friction (damping), so the net force is just: F = kx = ma = m d2 x dt 2, if define ω 2 0 = k/m, then d 2 x dt 2 + ω 0 2 x = 0 the general solution x = ACos(ω 0 t + φ), The kinetic energy is T = 1 2 m dx dt 2 = 1 2 ka2 sin 2 (ω 0 t + φ), the potential energy is U = 1 2 kx2 = 1 2 ka2 cos 2 (ω 0 t + φ) the total energy of the system has the constant value E = 1 2 ka2.

7 IPT5340, Spring 08 p. 7/5 Quantum Harmonic Oscillator: 1D In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a potential V (x) = 1 2 mω2 x 2. In classical mechanics, mω 2 = k is called the spring stiffness coefficient or force constant, and ω the circular frequency. The Hamiltonian of the particle is: H = p2 2m mω2 x 2 where x is the position operator, and p is the momentum operator p = i d. The first term dx represents the kinetic energy of the particle, and the second term represents the potential energy in which it resides.

8 IPT5340, Spring 08 p. 8/5 Maxwell s equations in Free space Faraday s law: Ampére s law: E = t B, H = t D, Gauss s law for the electric field: D = 0, Gauss s law for the magnetic field: B = 0, the constitutive relation: B = µ 0 H and D = ǫ 0 E.

9 IPT5340, Spring 08 p. 9/5 Plane electromagnetic waves Maxwell s equations in free space, there is vacuum, no free charges, no currents, J = ρ = 0, both E and B satisfy wave equation, 2 E = ǫ 0 µ 0 2 E t 2, we can use the solutions of wave optics, E(r, t) = E 0 exp(iωt)exp( ik r), B(r, t) = B 0 exp(iωt)exp( ik r),

10 IPT5340, Spring 08 p. 10/5 Mode Expansion of the Field A single-mode field, polarized along the x-direction, in the cavity: E(r, t) = ˆxE x (z, t) = X j ( 2m jω 2 j V ǫ 0 ) 1/2 q j (t)sin(k j z), where k = ω/c, ω j = c(jπ/l), j = 1,2,..., V is the effective volume of the cavity, and q(t) is the normal mode amplitude with the dimension of a length (acts as a canonical position, and p j = m j q j is the canonical momentum). the magnetic field in the cavity: 2ωj 2 H(r, t) = ŷh y (z, t) = (m j ) V ǫ 1/2 ( q j(t)ǫ 0 )Cos(k 0 k j z), j the classical Hamiltonian for the field: H = 1 Z dv [ǫ 0 Ex 2 + µ 0 Hy], 2 2 V = 1 X [m j ωmq 2 j m j q j 2 ] = 1 2 j X [m j ωmq 2 j 2 + p2 j ]. m j j

11 IPT5340, Spring 08 p. 11/5 Quantization of the Electromagnetic Field Like simple harmonic oscillator, Ĥ = p2 2m kx2, where [ˆx, ˆp] = i, For EM field, Ĥ = 1 2 Pj [m jω 2 mq 2 j + p2 j m j ], where [ˆq i, ˆp j ] = i δ ij, annihilation and creation operators: â j e iω jt = â j eiω jt = 1 p 2mj ω j (m j ω j ˆq j + iˆp j ), 1 p 2mj ω j (m j ω j ˆq j iˆp j ), the Hamiltonian for EM fields becomes: Ĥ = P j ω j(â jâj ), the electric and magnetic fields become, Ê x (z, t) = X j ( ω j ǫ 0 V )1/2 [â j e iω jt + â j eiω jt ]Sin(k j z), Ĥ y (z, t) = iǫ 0 c X j ( ω j ǫ 0 V )1/2 [â j e iω jt â j eiω jt ]Cos(k j z),

12 IPT5340, Spring 08 p. 12/5 Quantization of EM fields the Hamiltonian for EM fields becomes: Ĥ = P j ω j(â jâj ), the electric and magnetic fields become, Ê x (z, t) = X j = X j ( ω j ǫ 0 V )1/2 [â j e iω jt + â j eiω jt ]sin(k j z), c j [â 1j cos ω j t + â 2j sin ω j t]u j (r),

13 IPT5340, Spring 08 p. 13/5 Simple Harmonic Oscillator in Schrödinger picture one-dimensional harmonic oscillator, Ĥ = p2 2m kx2, Schrödinger equation, d 2 2m ψ(x) + dx2 2 [E 1 2 kx2 ]ψ(x) = 0, with dimensionless coordinates ξ = p mω/ x and dimensionless quantity ǫ = 2E/ ω, we have d 2 dξ 2 ψ(x) + [ǫ ξ2 ]ψ(x) = 0, which has Hermite-Gaussian solutions, ψ(ξ) = H n (ξ)e ξ2 /2, E = 1 2 ωǫ = ω(n ), where n = 0,1, 2,... Ch. 7 in Quantum Mechanics, by A. Goswami. Ch. 2 in Modern Quantum Mechanics, by J. Sakurai.

14 IPT5340, Spring 08 p. 14/5 which has Hermite-Gaussian solutions, Quantum Harmonic Oscillator d 2 dξ 2 ψ(x) + [ǫ ξ2 ]ψ(x) = 0, ψ(ξ) = H n (ξ)e ξ2 /2, E = 1 2 ωǫ = ω(n ), where n = 0,1, 2,...

15 IPT5340, Spring 08 p. 15/5 Simple Harmonic Oscillator: operator method one-dimensional harmonic oscillator, Ĥ = p2 2m kx2, where [ˆx, ˆp] = i define annihilation operator (destruction, lowering, or step-down operators): â = p mω/2 ˆx + iˆp/ 2m ω. define creation operator (raising, or step-up operators): â = p mω/2 ˆx iˆp/ 2m ω. note that â and â are not hermitian operators, but (â ) = â. the commutation relation for â and â is [â, â ] = 1. the oscillator Hamiltonian can be written as, Ĥ = ω(â â ) = ω( ˆN ), where ˆN is called the number operator, which is hermitian.

16 IPT5340, Spring 08 p. 16/5 Simple Harmonic Oscillator: operator method the number operator, ˆN = â â, [Ĥ, â] = ωâ, and [Ĥ, â ] = ωâ. the eigen-energy of the system, Ĥ Ψ = E Ψ, then Ĥâ Ψ = (E ω)â Ψ, Ĥâ Ψ = (E + ω)â Ψ. for any hermitian operator, Ψ ˆQ 2 Ψ = ˆQΨ ˆQΨ 0. thus Ψ Ĥ Ψ 0. ground state (lowest energy state), â Ψ 0 = 0. energy of the ground state, Ĥ Ψ 0 = 1 2 ω Ψ 0. excited state, Ĥ Ψ n = Ĥ(â ) n Ψ 0 = ω(n )(â ) n Ψ 0. eigen-energy for excited state, E n = (n ) ω.

17 IPT5340, Spring 08 p. 17/5 Simple Harmonic Oscillator: operator method normalization of the eigenstates, (â ) n Ψ 0 = c n Ψ n, where c n = n. â Ψ n = n Ψ n 1, â Ψ n = n + 1 Ψ n+1, x-representation, Ψ n (x) = x Ψ n. ground state, x â Ψ 0 = 0, i.e. r mω [ 2 x + 1 d 2m ω dx ]Ψ 0(x) = 0, define a dimensionless variable ξ = p mω/ x, we obtain (ξ + d dξ )Ψ 0 = 0, with the solution Ψ 0 (ξ) = c 0 exp( ξ 2 /2).

18 IPT5340, Spring 08 p. 18/5 brain-storms Damped harmonic oscillator: d2 x dt 2 + b dx m dt + ω 0 2 x = 0, where b is an experimentally determined damping constant satisfying the relationship F = bv. An example of a system obeying this equation would be a weighted spring underwater if the damping force exerted by the water is assumed to be linearly proportional to v. Mode expansion of the field in other bases, e.x. spherical wave: E(r) = A r r 0 exp( ik r r 0 ), How to quantize fields?

19 IPT5340, Spring 08 p. 19/5 Postulates of Quantum Mechanics Postulate 1: An isolated quantum system is described by a vector in a Hilbert space. Two vectors differing only by a multiplying constant represent the same physical state. quantum state: Ψ = P i α i ψ i, completeness: P i ψ i ψ i = I, probability interpretation (projection): Ψ(x) = x Ψ, operator: Â Ψ = Φ, representation: φ Â ψ, adjoint of Â: φ Â ψ = ψ Â φ, hermitian operator: Ĥ = Ĥ, unitary operator: ÛÛ = Û Û = I. Ch. 1-5 in The Principles of Quantum Mechanics, by P. Dirac. Ch. 1 in Mathematical Methods of Quantum Optics, by R. Puri.

20 IPT5340, Spring 08 p. 20/5 Operators For a unitary operator, ψ i ψ j = ψ i Û Ûψ j, the set of states the scalar product. Û ψ preserves Û can be represented as Û = exp(iĥ) if Ĥ is hermitian. normal operator: [Â,  ] = 0, the eigenstates of only a normal operator are orthonormal. i.e. hermitian and unitary operators are normal operators. The sum of the diagonal elements φ  ψ is call the trace of Â, Tr(Â) = X i φ i  φ i, The value of the trace of an operator is independent of the basis. The eigenvalues of a hermitian operator are real, Ĥ Ψ = λ Ψ, where λ is real. If  and ˆB do not commute then they do not admit a common set of eigenvectors.

21 IPT5340, Spring 08 p. 21/5 Postulates of Quantum Mechanics Postulate 2: To each dynamical variable there corresponds a unique hermitian operator. Postulate 3: If  and ˆB are hermitian operators corresponding to classical dynamical variables a and b, then the commutator of  and ˆB is given by [Â, ˆB]  ˆB ˆB = i {a, b}, where {a, b} is the classical Poisson bracket. Postulate 4: Each act of measurement of an observable  of a system in state Ψ collapses the system to an eigenstate ψ i of  with probability φ i Ψ 2. The average or the expectation value of  is given by  = X i λ i φ i Ψ 2 = Ψ Â Ψ, where λ i is the eigenvalue of  corresponding to the eigenstate ψ i.

22 IPT5340, Spring 08 p. 22/5 Uncertainty relation Non-commuting observable do not admit common eigenvectors. Non-commuting observables can not have definite values simultaneously. Simultaneous measurement of non-commuting observables to an arbitrary degree of accuracy is thus incompatible. variance:  2 = Ψ (  ) 2 Ψ = Ψ Â 2 Ψ Ψ Â Ψ 2. A 2 B [ ˆF 2 + Ĉ 2 ], where [Â, ˆB] = iĉ, and ˆF =  ˆB + ˆB 2  ˆB. Take the operators  = ˆq (position) and ˆB = ˆp (momentum) for a free particle, [ˆq, ˆp] = i ˆq 2 ˆp

23 IPT5340, Spring 08 p. 23/5 Uncertainty relation Schwarz inequality: φ φ ψ ψ φ ψ ψ φ. Equality holds if and only if the two states are linear dependent, ψ = λ φ, where λ is a complex number. uncertainty relation, A 2 B [ ˆF 2 + Ĉ 2 ], where [Â, ˆB] = iĉ, and ˆF =  ˆB + ˆB 2  ˆB. the operator ˆF is a measure of correlations between  and ˆB. define two states, ψ 1 = [  ] ψ, ψ 2 = [ ˆB ˆB ] ψ, the uncertainty product is minimum, i.e. ψ 1 = iλ ψ 2, [ + iλ ˆB] ψ = [  + iλ ˆB ] ψ = z ψ. the state ψ is a minimum uncertainty state.

24 IPT5340, Spring 08 p. 24/5 Uncertainty relation if Re(λ) = 0,  + iλ ˆB is a normal operator, which have orthonormal eigenstates. the variances,  2 = iλ 2 [ ˆF + i Ĉ ], ˆB 2 = i 2λ [ ˆF i Ĉ ], set λ = λ r + iλ i, Â2 = 1 2 [λ i ˆF + λ r Ĉ ], ˆB 2 = 1 λ 2 Â2, λ i Ĉ λ r ˆF = 0. if λ = 1, then  2 = ˆB 2, equal variance minimum uncertainty states. if λ = 1 along with λ i = 0, then  2 = ˆB 2 and ˆF = 0, uncorrelated equal variance minimum uncertainty states. if λ r 0, then ˆF = λ i λ Ĉ, Â2 = λ 2 r 2λ Ĉ, ˆB 2 = 1 r 2λ Ĉ. r If Ĉ is a positive operator then the minimum uncertainty states exist only if λ r > 0.

25 IPT5340, Spring 08 p. 25/5 Momentum as a generator of Translation For an infinitesimal translation by dx, and the operator that does the job by T (dx), T (dx) x = x + dx, the infinitesimal translation should be unitary, T (dx)t (dx) = 1, two successive infinitesimal translations, T (dx 1 )T (dx 2 ) = T (dx 1 + dx 2 ), a translation in the opposite direction, T (dx 1 ) = T 1 (dx), identity operation, dx 0, then lim dx 0 T (dx) = 1, define a Hermitian operator, T (dx) = exp( i ˆK dx) 1 i ˆK dx, Ch. 2 in Modern Quantum Mechanics, by J. Sakurai.

26 IPT5340, Spring 08 p. 26/5 Momentum as a generator of Translation define a Hermitian operator, T (dx) = exp( i ˆK dx) 1 i ˆK dx, we have the communication relation, [ˆx, ] = dx, or [ˆx i, ˆK j ] = iδ ij, L. De Brogie s relation, define ˆK = ˆp/, then 2π λ = p, [ˆx i, ˆp j ] = i δ ij, Ch. 2 in Modern Quantum Mechanics, by J. Sakurai.

27 IPT5340, Spring 08 p. 27/5 Momentum Operator in the Position basis the definition of momentum as the generator of infinitesimal translations, (1 iˆp x ) α = = = = Z Z Z Z dxt ( x) x x α dx x + x x α dx x x x α dx x ( x α x x x α ) comparison of both sides, ˆp α = Z dx x ( i x x α ), or x ˆp α = i x x α

28 IPT5340, Spring 08 p. 28/5 Uncertainty relation for ˆq and ˆp take the operators  = ˆq (position) and ˆB = ˆp (momentum) for a free particle, [ˆq, ˆp] = i ˆq 2 ˆp define two states, ψ 1 = [  ] ψ ˆα ψ, ψ 2 = [ ˆB ˆB ] ψ ˆβ ψ. for uncorrelated minimum uncertainty states, ˆα ψ = iλˆβ ψ, ψ ˆαˆβ + ˆβˆα ψ = 0, where λ is a real number. if  = ˆq and ˆB = ˆp, we have (ˆq ˆq ) ψ = iλ(ˆp ˆp ) ψ. the wavefunction in the q-basis is, i.e. ˆp = i / q, ψ(q) = q ψ = 1 exp[i ˆp q (2π ˆq 2 ) 1/4 (q ˆq )2 4 ˆq 2 ], in the p-basis, ψ(p) = p ψ = 1 exp[ i (p ˆp )2 ( ˆq (p ˆp ) (2π ˆp 2 ) 1/4 4 ˆp 2 ].

29 IPT5340, Spring 08 p. 29/5 Minimum Uncertainty State (ˆq ˆq ) ψ = iλ(ˆp ˆp ) ψ if we define λ = e 2r, then (e rˆq + ie rˆp) ψ = (e r ˆq + ie r ˆp ) ψ, the minimum uncertainty state is defined as an eigenstate of a non-hermitian operator e rˆq + ie rˆp with a c-number eigenvalue e r ˆq + ie r ˆp. the variances of ˆq and ˆp are ˆq 2 = 2 e 2r, ˆp 2 = 2 e2r. here r is referred as the squeezing parameter.

30 IPT5340, Spring 08 p. 30/5 Gaussian Wave Packets in the x-space, 1 Ψ(x) = x Ψ = [ π 1/4 d ]exp[ikx x2 2d 2 ], which is a plane wave with wave number k and width d. the expectation value of ˆX is zero for symmetry, ˆX = Z dx Ψ x ˆX x Ψ = 0. variation of ˆX, ˆX 2 = d2 2. the expectation value of ˆP, ˆP = k, i.e. x ˆP Ψ = i x x Ψ. variation of ˆP, ˆP 2 = 2 2d 2. the Heisenberg uncertainty product is, ˆX 2 ˆP 2 = 2 4. a Gaussian wave packet is called a minimum uncertainty wave packet.

31 IPT5340, Spring 08 p. 31/5 Phase diagram for EM waves Electromagnetic waves can be represented by Ê(t) = E 0 [ ˆX 1 sin(ωt) ˆX 2 cos(ωt)] where ˆX 1 = amplitude quadrature ˆX 2 = phase quadrature

32 IPT5340, Spring 08 p. 32/5 Quadrature operators the electric and magnetic fields become, Ê x (z, t) = X j = X j ( ω j ǫ 0 V )1/2 [â j e iω jt + â j eiω jt ]sin(k j z), c j [â 1j cos ω j t + â 2j sin ω j t]u j (r), note that â and â are not hermitian operators, but (â ) = â. â 1 = 1 2 (â + â ) and â 2 = 1 2i (â â ) are two Hermitian (quadrature) operators. the commutation relation for â and â is [â, â ] = 1, the commutation relation for â and â is [â 1, â 2 ] = i 2, and â 2 1 â

33 Phase diagram for coherent states IPT5340, Spring 08 p. 33/5

34 IPT5340, Spring 08 p. 34/5 Coherent and Squeezed States Uncertainty Principle: ˆX 1 ˆX Coherent states: ˆX 1 = ˆX 2 = 1, 2. Amplitude squeezed states: ˆX 1 < 1, 3. Phase squeezed states: ˆX 2 < 1, 4. Quadrature squeezed states.

35 IPT5340, Spring 08 p. 35/5 Vacuum, Coherent, and Squeezed states vacuum coherent squeezed-vacuum amp-squeezed phase-squeezed quad-squeezed

36 IPT5340, Spring 08 p. 36/5 Generations of Squeezed States Nonlinear optics: Courtesy of P. K. Lam

37 IPT5340, Spring 08 p. 37/5 Generation and Detection of Squeezed Vacuum 1. Balanced Sagnac Loop (to cancel the mean field), 2. Homodyne Detection. M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153(1991).

38 IPT5340, Spring 08 p. 38/5 Schrödinger equation Postulate 5: The time evolution of a state Ψ is governed by the Schrödinger equation, i d Ψ(t) = Ĥ(t) Ψ(t), dt where Ĥ(t) is the Hamiltonian which is a hermitian operator associated with the total energy of the system. The solution of the Schrödinger equation is, Ψ(t) = T exp[ i Z t t 0 dτĥ(τ)] Ψ(0) ÛS(t, t 0 ) Ψ(t 0 ), where ( T) is the time-ordering operator. Schrödinger picture: Ψ(r, t) = X i α i (t) ψ i (r).

39 IPT5340, Spring 08 p. 39/5 Time Evolution of a Minimum Uncertainty State the Hamiltonian for a free particle, Ĥ = ˆp2 2m, then Û = exp( i ˆp 2 2m t). the Schrödinger wavefunction, Ψ(q, t) = q Û Ψ(0) = Z = dp p Ψ(p,0)exp( i where q = /2 ˆp 2 1/2, and q p = 1 2π exp( ipq ). p 2 2m t), 1 (2π) 1/4 ( q + i t/2m q) 1/2 exp[ q 2 4( q) 2 + 2i t/m ], even though the momentum uncertainty ˆp 2 is preserved, the position uncertainty increases as time develops, ˆq 2 (t) = ( ˆq) t 2 4m 2 ( q) 2

40 IPT5340, Spring 08 p. 40/5 Gaussian Optics Wave equation: In free space, the vector potential, A, is defined as A(r, t) = nψ(x, y, z)e jωt, which obeys the vector wave equation, 2 ψ + k 2 ψ = 0. The paraxial wave equation: ψ(x, y, z) = u(x, y, z)e jkz, one obtains where T ˆx x + ŷ y. 2 T u 2jk u z = 0, This solution is proportional to the impulse response function (Fresnel kernel), i.e. 2 T h(x, y, z) 2jk h z = 0. h(x, y, z) = j λz e jk[(x2 +y 2 )/2z],

41 IPT5340, Spring 08 p. 41/5 Gaussian Optics The solution of the scalar paraxial wave equation is, u 00 (x, y, z) = 2 exp(jφ)exp( x2 + y 2 πw w 2 )exp[ jk 2R (x2 + y 2 ], beam width w 2 (z) = 2b k z2 (1 + b 2 = w0 2 λz [1 + ( πw0 2 ) 2 ], radius of phase front 1 R(z) = z z 2 +b 2 = z z 2 +(πw 2 0 /λ)2, phasedelay tan φ = z b = z πw 2 0 /λ, with the minimum beam radius w 0 = 2bk x u_ x z z

42 IPT5340, Spring 08 p. 42/5 Heisenberg equation The solution of the Schrödinger equation is, Ψ(t) = T exp[ i R t t dτĥ(τ)] Ψ(0) ÛS(t, t 0 ) Ψ(t 0 ). 0 The quantities of physical interest are the expectation values of operators, Ψ(t) Â Ψ(t) = Ψ(t 0) Â(t) Ψ(t 0), where Â(t) = Û S (t, t 0)ÂÛS(t, t 0 ). The time-dependent operator Â(t) evolves according to the Heisenberg equation, i d Â(t) = [Â, Ĥ(t)]. dt Schrödinger picture: time evolution of the states. Heisenberg picture: time evolution of the operators.

43 IPT5340, Spring 08 p. 43/5 Interaction picture Consider a system described by Ψ(t) evolving under the action of a hamiltonian Ĥ(t) decomposable as, where Ĥ0 is time-independent. Ĥ(t) = Ĥ 0 + Ĥ 1 (t), Define Ψ I (t) = exp(iĥ0t/ ) Ψ(t), then Ψ I (t) evolves accords to i d dt Ψ I(t) = ĤI(t) Ψ I (t), where Ĥ I (t) = exp(iĥ0t/ )Ĥ1(t)exp( iĥ0t/ ). The evolution is in the interaction picture generated by Ĥ0.

44 IPT5340, Spring 08 p. 44/5 Paradoxes of Quantum Theory Geometric phase Measurement theory Schrödinger s Cat paradox Einstein-Podolosky-Rosen paradox Local Hidden Variables theory

45 IPT5340, Spring 08 p. 45/5 Quantum Zeno effect (watchdog effect) multi-time joint probability: P({ φ i, t i }), the probability that a system in a state φ 0 (t 0 ) at t 0 is found in the state φ i at t i, where i = 1,..., n. at t 1 : the state is ÛS(t 1, t 0 ) φ 0 (t 0 ). projection on φ 1 is φ 1 (t 1 ) = φ 1 φ 1 Û S (t 1, t 0 ) φ 0 (t 0 ). the sate φ 1 (t 1 ) then evolves till time t 2 to ÛS(t 2, t 1 ) φ 1 (t 1 ), with the projection, φ 2 (t 2 ) = φ 2 φ 2 ÛS(t 2, t 1 ) φ 1 (t 1 ). continuing till time t n, ny P({ φ i, t i }) = φ i ÛS(t i, t i 1 ) φ i 1 2. i=1

46 IPT5340, Spring 08 p. 46/5 Quantum Zeno effect (watchdog effect) consider a time-independent hamiltonian, ÛS(t i, t j ) = exp[ iĥ(t i t j )/ ]. let the observation be spaced at equal time intervals, t i t i 1 = t/n. the probability that at each time t i the system is observed in its initial state φ 0 is, P({ φ 0, t i }) = φ 0 exp[ iĥt/n ] φ 0 2n. let t/n 1, φ 0 exp[ iĥt/n ] φ ( t n )2 Ĥ2, where Ĥ2 = φ 0 Ĥ2 φ 0 φ 0 Ĥ φ 0 2.

47 IPT5340, Spring 08 p. 47/5 Quantum Zeno effect (watchdog effect) the joint probability for n equally spaced observations becomes, P({ φ 0, t i }) = [1 ( t n )2 Ĥ 2 ] n. for unobserved in between, the probability is, P({ φ 0, t}) = 1 ( t2 2 ) Ĥ2. the probability of finding the system in its initial state at a given time is increased if it is observed repeatedly at intermediate times. for n 1, P({ φ 0, t i }) = [1 ( t n )2 Ĥ2 ] n exp[ t 2 Ĥ2 /n 2 ], the system under observation does not evolve. this effect was invoked to predict the inhibition of decay of an unstable system.

48 IPT5340, Spring 08 p. 48/5 Time-dependent perturbation theory with the interaction picture, Ĥ = Ĥ0 + Ĥ1. the state, Ψ(r, t) = P n C n(t)u n (r)e iω nt with the energy eigenvalue Ĥ 0 u n (r) = ω n u n (r). the wavefunction has the initial value, Ψ(r, 0) = u i (r), i.e. C i (0) = 1, C n i = 0. the equation of motion for the probability amplitude C n (t) is, C n (t) = i X n Ĥ1 m e iωnmt C m (t), m C n (1) (t) = i 1 n Ĥ1 i e iω nit. if Ĥ1 = V 0 time independent, we have C n (t) C n (1) (t) = i 1 n Ĥ 1 i eiω nit 1 iω ni = i 1 n Ĥ 1 i e iω nit/2 sin(ω nit/2) ω ni /2 Ch. 3 in Elements of Quantum Optics, by P. Meystre and M. Sargent III. Ch. 5 in Modern Quantum Mechanics, by J. Sakurai.

49 IPT5340, Spring 08 p. 49/5 Rotational-Wave Approximation if Ĥ1 = V 0 cos νt, we have C n (t) C (1) n (t) = i V ni [ei(ωni+ν)t 1 2 i(ω ni + ν) + ei(ω ni ν)t 1 i(ω ni ν) ], where V ni = n Ĥ1 i. if near resonance ω ni ν, we can neglect the terms with ω ni + ν. This is called the rotational-wave approximation. making the rotational-wave approximation, C n (1) 2 = V ni 2 sin 2 [(ω ni ν)t/2] 4 2 (ω ni ν) 2. /4 we have the same transition probability as the dc case, provided we substitute ω ni ν for ω ni.

50 IPT5340, Spring 08 p. 50/5 Fermi-Golden rule the total transition probability from an initial state to the final state is, P T Z D(ω) C (1) n 2 dω, where D(ω) is the density of state factor. Fermi-Golden rule, P T = Z V (ω) 2 dωd(ω) 4 2 t 2 sin2 [(ω ni ν)t/2] [(ω ni ν)t/2] 2. consider resonance condition ω = ν, P T D(ν) = Z V (ν) t 2 π 2 2 D(ν) V (ν) 2 t. dω sin2 [(ω ni ν)t/2] [(ω ni ν)t/2] 2, the transition rate, Γ = dp T dt in time. = d dt C(1) n 2 = π 2 2 D(ν) V (ν) 2, which is a constant

51 Casimir effect IPT5340, Spring 08 p. 51/5